Given the importance of linear models in statistical theory and experimental research, a good understanding of their fundamental principles and theory is essential. Supported by a large number of examples, Linear Model Methodology provides a strong foundation in the theory of linear models and explores the latest developments in data analysis.
After presenting the historical evolution of certain methods and techniques used in linear models, the book reviews vector spaces and linear transformations and discusses the basic concepts and results of matrix algebra that are relevant to the study of linear models. Although mainly focused on classical linear models, the next several chapters also explore recent techniques for solving well-known problems that pertain to the distribution and independence of quadratic forms, the analysis of estimable linear functions and contrasts, and the general treatment of balanced random and mixed-effects models. The author then covers more contemporary topics in linear models, including the adequacy of Satterthwaite’s approximation, unbalanced fixed- and mixed-effects models, heteroscedastic linear models, response surface models with random effects, and linear multiresponse models. The final chapter introduces generalized linear models, which represent an extension of classical linear models.
Linear models provide the groundwork for analysis of variance, regression analysis, response surface methodology, variance components analysis, and more, making it necessary to understand the theory behind linear modeling. Reflecting advances made in the last thirty years, this book offers a rigorous development of the theory underlying linear models.
Linear Models: Some Historical Perspectives
The Invention of Least Squares
The Gauss–Markov Theorem
Estimability
Maximum Likelihood Estimation
Analysis of Variance (ANOVA)
Quadratic Forms and Craig’s Theorem
The Role of Matrix Algebra
The Geometric Approach
Basic Elements of Linear Algebra
Introduction
Vector Spaces
Vector Subspaces
Bases and Dimensions of Vector Spaces
Linear Transformations
Basic Concepts in Matrix Algebra
Introduction and Notation
Some Particular Types of Matrices
Basic Matrix Operations
Partitioned Matrices
Determinants
The Rank of a Matrix
The Inverse of a Matrix
Eigenvalues and Eigenvectors
Idempotent and Orthogonal Matrices
Quadratic Forms
Decomposition Theorems
Some Matrix Inequalities
Function of Matrices
Matrix Differentiation
The Multivariate Normal Distribution
History of the Normal Distribution
The Univariate Normal Distribution
The Multivariate Normal Distribution
The Moment Generating Function
Conditional Distribution
The Singular Multivariate Normal Distribution
Related Distributions
Examples and Additional Results
Quadratic Forms in Normal Variables
The Moment Generating Function
Distribution of Quadratic Forms
Independence of Quadratic Forms
Independence of Linear and Quadratic Forms
Independence and Chi-Squaredness of Several Quadratic Forms
Computing the Distribution of Quadratic Forms
Appendix
Full-Rank Linear Models
Least-Squares Estimation
Properties of Ordinary Least-Squares Estimation
Generalized Least-Squares Estimation
Least-Squares Estimation under Linear Restrictions on β
Maximum Likelihood Estimation
Inference Concerning β
Examples and Applications
Less-Than-Full-Rank Linear Models
Parameter Estimation
Some Distributional Properties
Reparameterized Model
Estimable Linear Functions
Simultaneous Confidence Intervals on Estimable Linear Functions
Simultaneous Confidence Intervals on All Contrasts among the Means with Heterogeneous Group Variances
Further Results Concerning Contrasts and Estimable Linear Functions
Balanced Linear Models
Notation and Definitions
The General Balanced Linear Model
Properties of Balanced Models
Balanced Mixed Models
Complete and Sufficient Statistics
ANOVA Estimation of Variance Components
Confidence Intervals on Continuous Functions of the Variance Components
Confidence Intervals on Ratios of Variance Components
The Adequacy of Satterthwaite’s Approximation
Satterthwaite’s Approximation
Adequacy of Satterthwaite’s Approximation
Measuring the Closeness of Satterthwaite’s Approximation
Examples
Appendix
Unbalanced Fixed-Effects Models
The R-Notation
Two-Way Models without Interaction
Two-Way Models with Interaction
Higher-Order Models
A Numerical Example
The Method of Unweighted Means
Unbalanced Random and Mixed Models
Estimation of Variance Components
Estimation of Estimable Linear Functions
Inference Concerning the Random One-Way Model
Inference Concerning the Random Two-Way Model
Exact Tests for Random Higher-Order Models
Inference Concerning the Mixed Two-Way Model
Inference Concerning the Random Two-Fold Nested Model
Inference Concerning the Mixed Two-Fold Nested Model
Inference Concerning the General Mixed Linear Model
Appendix
Additional Topics in Linear Models
Heteroscedastic Linear Models
The Random One-Way Model with Heterogeneous Error Variances
A Mixed Two-Fold Nested Model with Heteroscedastic Random Effects
Response Surface Models
Response Surface Models with Random Block Effects
Linear Multiresponse Models
Generalized Linear Models
Introduction
The Exponential Family
Estimation of Parameters
Goodness of Fit
Hypothesis Testing
Confidence Intervals
Gamma-Distributed Response
Bibliography
Index
Exercises appear at the end of each chapter, except for Chapter 1.
Biography
André I. Khuri is a Professor Emeritus in the Department of Statistics at the University of Florida in Gainesville.
This is a comprehensive and up-to-date textbook on the theory of linear models. … Every chapter besides the first historical one contains many exercises … . There is also a huge bibliography. The textbook represents an important source for all researchers and lectures in linear models.
—Hilmar Drygas, Zentralblatt MATH, 2012The outstanding book, written by a prominent researcher and author, presents a wealth of materials on linear models in Chapters 1 though 12 and includes materials on generalized linear models in the last chapter. The material on linear models is an amazing collection of important topics that would benefit researchers, teachers, students, and practitioners and has added value to the book. Many illustrative examples are presented with SAS codes. The examples are practically important and thoughtfully chosen. Exercises at the end of Chapters 2-13 are excellent, and some are valuable resources for the researchers in this area. … The author has to be commended for his success in executing this so elegantly.
—Subir Ghosh, Journal of Quality Technology, Vol. 44, January 2012This book provides a very well-written and rigorous account of the theory of linear models. … In sum, this is a carefully written and reliable book that reflects the experience of the author in teaching graduate level courses on linear models. I will certainly add it to the list of reference textbooks for the graduate one-quarter course on linear model theory taught at UC Santa Cruz.
—Raquel Prado, Journal of the American Statistical Association, September 2011, Vol. 106This text is a possible choice for a second course in linear model theory.
—David J. Olive, Technometrics, May 2011The material is well chosen and well organized, and includes many results that are not found in other textbooks. … Throughout the book, the presentation is very clear and well organized, with a focus on mathematical developments. Most results are stated with proofs, some material is based on the author’s own contributions to the field. Generally, many important special cases are treated in detail, which will make the book also highly useful as a reference. There are also many worked-out examples from different subject areas to illustrate the methods. Later chapters also include some instructions on how to use the methods in SAS. Furthermore, there are lots of exercises at the end of each chapter. … The book is very accessible and encompassing … the book will be an excellent choice both as a text and as a reference book.
—T. Mildenberger, Statistical Papers, April 2011The material on which this book is based has been taught in a couple of courses at the University of Florida for about 20 years and the author’s skills and experience in doing this are superbly represented in this fine text. … there are numerous exercises that reinforce both the theoretical and the practical aspects of regression… This is an excellent, reliable, and comprehensive text.
—International Statistical Review (2010), 78This book provides a thorough overview which is similar to other available texts but in a very different way. The choice of topics covered, their organization and presentation are the unique features that distinguish this book. … This book is well structured as a textbook as well as a reference with every chapter explaining the definitions, principles and methods of the subject matter illustrated by data-based examples with the details on use of SAS software, wherever possible. … the topics that are covered in Chapters 7–12 are not generally found in a single book. … The book would make an excellent textbook for a course on linear models at masters and graduate levels. Moreover, some parts of the book can also be a part of a course on analysis of variance. Overall, the book is a valuable reference for those involved in research and teaching in this area.
—Journal of the Royal Statistical Society, Series A, 2010
